×

zbMATH — the first resource for mathematics

Properties of direct summands of modules. (English) Zbl 0659.16016
L. Fuchs posed [in his “Infinite abelian groups” I (1970; Zbl 0209.055)] the problem of characterizing the abelian groups in which the intersection of each pair of direct summands is a direct summand. This property (called the summand intersection property or SIP) has been studied, for arbitrary modules, by G. Wilson [Commun. Algebra 14, 21-38 (1986; Zbl 0592.13008)], who also characterized modules with SIP over principal ideal domains. In this paper, modules M are studied with the property that the sum of any pair of direct summands of M is a direct summand of M (summand property or SSP). Characterizations of rings are obtained in terms of the SSP of their modules, and, for a module M, the SSP (or the SIP) of M is studied through properties of \(End_ R(M)\). Furthermore the author studies the conditions when a given module, which can be written as a direct sum of indecomposable modules, has the SSP (or the SIP or both). Some particular cases are investigated.
Reviewer: F.Loonstra

MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S50 Endomorphism rings; matrix rings
16W50 Graded rings and modules (associative rings and algebras)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson F.W., Rings and categories of modules (1974) · Zbl 0301.16001
[2] Faith C., Algebra II:Ring Theory (1976)
[3] Fuchs L., Infinite abelian groups 1 (1970)
[4] DOI: 10.1080/00927878508823148 · Zbl 0549.16014
[5] DOI: 10.1093/qmath/22.2.173 · Zbl 0215.09102
[6] DOI: 10.1007/BF01113346 · Zbl 0197.30901
[7] DOI: 10.1080/00927877608822103 · Zbl 0322.16011
[8] DOI: 10.4153/CJM-1976-109-2 · Zbl 0317.16005
[9] DOI: 10.1080/00927877708822166 · Zbl 0358.16002
[10] DOI: 10.1090/S0002-9947-1971-0274511-2
[11] DOI: 10.1080/00927878608823297 · Zbl 0592.13008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.